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The Luhn algorithm or Luhn formula, also known as the "modulus 10" or "mod 10" algorithm, named after its creator, IBM scientist Hans Peter Luhn, is a simple check digit formula used to validate a variety of identification numbers. It is described in U.S. Patent No. 2,950,048, granted on August 23, 1960.
If a + k ≡ b + k (mod m), where k is any integer, then a ≡ b (mod m). If k a ≡ k b (mod m) and k is coprime with m, then a ≡ b (mod m). If k a ≡ k b (mod k m) and k ≠ 0, then a ≡ b (mod m). The last rule can be used to move modular arithmetic into division. If b divides a, then (a/b) mod m = (a mod b m) / b .
In mathematics, the result of the modulo operation is an equivalence class, and any member of the class may be chosen as representative; however, the usual representative is the least positive residue, the smallest non-negative integer that belongs to that class (i.e., the remainder of the Euclidean division ). [2]
The gcd(4, 10) = 2 and 2 does not divide 5, but does divide 6. Since gcd(3, 10) = 1 , the linear congruence 3 x ≡ 1 (mod 10) will have solutions, that is, modular multiplicative inverses of 3 modulo 10 will exist.
The Montgomery forms of 7 and 15 are 70 mod 17 = 2 and 150 mod 17 = 14, respectively. Their product 28 is the input T to REDC, and since 28 < RN = 170, the assumptions of REDC are satisfied. To run REDC, set m to (28 mod 10) ⋅ 7 mod 10 = 196 mod 10 = 6. Then 28 + 6 ⋅ 17 = 130, so t = 13.
Integer multiplication respects the congruence classes, that is, a ≡ a' and b ≡ b' (mod n) implies ab ≡ a'b' (mod n). This implies that the multiplication is associative, commutative, and that the class of 1 is the unique multiplicative identity.
The multiplicative order of a number a modulo n is the order of a in the multiplicative group whose elements are the residues modulo n of the numbers coprime to n, and whose group operation is multiplication modulo n.
In mathematics, the term modulo ("with respect to a modulus of", the Latin ablative of modulus which itself means "a small measure") is often used to assert that two distinct mathematical objects can be regarded as equivalent—if their difference is accounted for by an additional factor.
Clearly, A ≠ G = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}. Let b 1 be an element of G\A; for instance, take b 1 = 2. Then, since 2×1 = 2, 2×5 = 10, 2×8 = 16 ≡ 3 (mod 13), 2×12 = 24 ≡ 11 (mod 13), we have A 1 = {2, 3, 10, 11}. Clearly, A∪A 1 ≠ G. Let b 2 be an element of G\(A∪A 1); for instance, take b 2 = 4. Then, since
The Luhn mod N algorithm is an extension to the Luhn algorithm (also known as mod 10 algorithm) that allows it to work with sequences of values in any even-numbered base. This can be useful when a check digit is required to validate an identification string composed of letters, a combination of letters and digits or any arbitrary set of N ...